What will happen if minimal polynomial co-incides with the characteristics polynomial? If $A$ is a $n \times n$ matrix such that it's minimal polynomial co-incides with the characteristics polynomial then can we claim that any $n-th$ degree polynomial which annihilates $A$ co-incides with the characteristics polynomial?
If the answer is 'yes' then how?I have thought about it fruitlessly.My thoughts are not well enough to understand this concept clearly.Please help me.
Thank you in advance.
 A: Let $P$ be the minimum polynomial of $M$, fuppose $Q$ is a polynomial with the same degree as $M$ that annihalates $M$.
Write $P$ as $\alpha Q+R$ were $R$ is of degree less than $P$ (we can do this with the division algorithm).
Notice that $0=P(M)=\alpha Q(M)+R(M)=0+R(M)$. This implies $R$ is the zero polynomial (since otherwise it would be a non-zero polynomial of degree less than $P$ that annihalates $M$. We conclude $Q=\alpha P$.
Your case is a particular case of this, you are just saying that $P$ has degree $n$.
A: Yes.  By definition, the minimal polynomial $m_A$ divides any other polynomial that annihilates $A$.  So, if some polynomial $f$ of equal degree annihilates $A$, then $m_A\mid f$, so $m_A = cf$ where $c$ is some constant.
A: There are other things that happen when the minimal polynomial and characteristic polynomial coincide; note that we demand both monic...
First, while there may be eigenvalues with multiplicity greater than one, nevertheless each eigenvalue occurs in a single Jordan block.
Second, if we call our matrix $A,$ then any matrix $B$ that commutes with $A,$ that is $AB=BA,$ is a polynomial in $A,$ of degree no larger than $n-1$ because of Cayley Hamilton, anyway
$$ B = a_0 I + a_1 A + a_2 A^2 + \cdots + a_{n-1} A^{n-1}. $$
The set of such $B$ makes a vector space, it is then dimension $n,$ which is very small. In comparison, the identity matrix commutes with all matrices, in that case dimension $n^2.$ Big.
Here's a simple example, one you can check with 2 by 2 and 3 by 3 matrices. If a diagonal matrix $D$ has $n$ different elements on the diagonal, then it commutes only with other diagonal matrices. These other diagonal matrices may have repetition, for example the identity matrix. 
